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Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)

  • Jairo Z. Gonçalves EMAIL logo
Published/Copyright: March 8, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 3

Abstract

Let k be a field, let 𝔄1 be the k-algebra k⁢[x1±1,…,xs±1] of Laurent polynomials in x1,…,xs, and let 𝔄2 be the k-algebra k⁢[x,y] of polynomials in the commutative indeterminates x and y. Let σ1 be the monomial k-automorphism of 𝔄1 given by A=(ai,j)∈G⁢Ls⁢(ℤ) and σ1⁢(xi)=∏j=1sxjai,j, 1≤i≤s, and let σ2∈Autk⁢(k⁢[x,y]). Let Di, 1≤i≤2, be the ring of fractions of the skew polynomial ring 𝔄i⁢[X;σi], and let Di∙ be its multiplicative group. Under a mild restriction for D1, and in general for D2, we show that Di∙, 1≤i≤2, contains a free subgroup. If i=1 and s=2, we show that a noncentral normal subgroup N of D1∙ contains a free subgroup.

Keywords: Division rings; free groups; normal subgroups
MSC 2010: 16K40; 20E05; 12E15

Communicated by Manfred Droste

Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award Identifier / Grant number: 301.205/2015-9

Funding source: Fundação de Amparo à Pesquisa do Estado de São Paulo

Award Identifier / Grant number: 2015/09162-9

Funding statement: The author was supported by Fapesp-Brazil, Projeto Tematico 2015/09162-9, and by Grant CNPq 301.205/2015-9.

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Received: 2017-11-29
Revised: 2019-02-16
Published Online: 2019-03-08
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
https://doi.org/10.1515/forum-2017-0248
Keywords for this article
Division rings; free groups; normal subgroups

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
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